Theorem raleqbidvv 3333

Description: Version of raleqbidv 3340 with additional disjoint variable conditions, not requiring ax-8 2148 nor df-clel 2840. (Contributed by BJ, 22-Sep-2024.)

Hypotheses

Ref	Expression
raleqbidvv.1	⊢ (𝜑 → 𝐴 = 𝐵)
raleqbidvv.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
raleqbidvv	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem raleqbidvv

Step	Hyp	Ref	Expression
1		raleqbidvv.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		raleqbidvv.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	raleqbidva 3331	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146  ∀wral 3081

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757  df-ral 3082  df-rex 3092

This theorem is used by:  raleqbi1dv  3335  dfttc4lem2  37101  gpg5nbgrvtx03star  48922  postcposALT  50422